Kuramoto Orientation Diffusion Models

TL;DR

This work introduces Kuramoto Orientation Diffusion Models, a nonlinear score-based generative framework that operates on periodic angular data by embedding forward synchronization dynamics inspired by the Kuramoto model. By coupling phase variables and incorporating a global reference phase, the forward diffusion organizes data into a low-entropy von Mises distribution, while the reverse diffusion desynchronizes through learned periodic scores to reconstruct diverse orientation patterns. The method uses wrapped Gaussian transitions and periodicity-aware neural networks to respect circular geometry, with global and local coupling variants yielding faster convergence and improved quality on orientation-dense datasets such as fingerprints and textures, and competitive results on general benchmarks. Overall, the approach showcases how biologically inspired synchronization priors can serve as effective priors for structured image generation and other angular data domains, enabling interpretable coarse-to-fine generation and improved sample quality with fewer diffusion steps.

Abstract

Orientation-rich images, such as fingerprints and textures, often exhibit coherent angular directional patterns that are challenging to model using standard generative approaches based on isotropic Euclidean diffusion. Motivated by the role of phase synchronization in biological systems, we propose a score-based generative model built on periodic domains by leveraging stochastic Kuramoto dynamics in the diffusion process. In neural and physical systems, Kuramoto models capture synchronization phenomena across coupled oscillators -- a behavior that we re-purpose here as an inductive bias for structured image generation. In our framework, the forward process performs \textit{synchronization} among phase variables through globally or locally coupled oscillator interactions and attraction to a global reference phase, gradually collapsing the data into a low-entropy von Mises distribution. The reverse process then performs \textit{desynchronization}, generating diverse patterns by reversing the dynamics with a learned score function. This approach enables structured destruction during forward diffusion and a hierarchical generation process that progressively refines global coherence into fine-scale details. We implement wrapped Gaussian transition kernels and periodicity-aware networks to account for the circular geometry. Our method achieves competitive results on general image benchmarks and significantly improves generation quality on orientation-dense datasets like fingerprints and textures. Ultimately, this work demonstrates the promise of biologically inspired synchronization dynamics as structured priors in generative modeling.

Figures (15)

• Figure 1: Governing stochastic differential equations (SDEs) and representative image samples from our globally and locally coupled Kuramoto orientation diffusion models. Pixels are mapped onto periodic domains as angular phase variables. In the globally coupled model, each pixel interacts with all other pixels via Kuramoto sinusoidal coupling (highlighted in red). The locally coupled variant corresponds to a similar SDE but restricts this sinusoidal coupling to a local neighborhood around each pixel. Unlike standard diffusion models, our approach introduces non-isotropic noise dynamics via pulling similar phases together, enabling a more structured destruction process. These dynamics help preserve the global structure in the early stages of diffusion (e.g., the overall shape of the bird), while allowing for faster adaptation to noise as the process progresses. The forward SDE synchronizes phase variables through oscillator interactions and a global reference phase. The reverse process desynchronizes these variables using learned score functions to synthesize images.
• Figure 2: Illustration of our Kuramoto orientation diffusion model. In the forward process (left-to-right), angular phase variables (colored circles) synchronize toward a low-entropy von Mises distribution, guided by attraction to a reference phase (white rectangle). The reverse process (right-to-left) uses learned score functions to desynchronize these phases.
• Figure 3: Samples generated by our Kuramoto orientation diffusion model on SOCOFing fingerprint and Brodatz texture datasets under varying denoising steps.
• Figure 4: Samples generated by our Kuramoto model on the ground terrain dataset.
• Figure 5: Samples generated by our Kuramoto model on CIFAR10 under varying denoising steps.